Multi-Page Printing

You have printed 0 times in the last 24 hours.
Your print count will reset on at .
You may print 0 more time(s) before then.
You may print a maximum of 0 pages at a time.

Number of pages to print:

NOTE: Printing will start on the current page.
Firefox users may need to click "Back" when printing completes.

Those Fascinating Numbers
Page133(153 of 451)

Those Fascinating Numbers 133 1 618 • the integer part131 of ee2 . 1 627 • the prime number which appears the most often as the 12th prime factor of an integer (see the number 199); • the smallest number n for which π(n) 3 j=1 (j − 1)!n logj n , this last expression representing the first three terms of the asymptotic expansion of Li(n): here π(1627) = 258 while n log n + n log2 n + 2n log3 n n=1627 ≈ 257.832 (see the number 73); • the ninth prime number p such that (3p − 1)/2 is itself a prime number (see the number 1 091). 1 634 • the second number which can be written as the sum of the fourth powers of its digits: 1 634 = 14 + 64 + 34 + 44; the others are 1, 8 208 and 9 474; • the sixth solution of σ(n) = σ(n + 1) (see the number 206). 1 638 • the fourth number which is neither perfect nor multi-perfect but whose har- monic mean is an integer (see the number 140); • the fourth solution of σ(n) n = 8 3 (see the number 1 488). 1 639 • the 1 000th square-free number (see the number 165). 131The interest for this number stems from a number theory result according to which the kth prime factor qk (n) of a number n is “usually” of the order of eek , in the sense that, for all ε 0 and any function ξ(n) which tends to +∞ as n → ∞, sup ξ(n)≤k≤ω(n) log log qk(n) − k 2k log k ≤ 1 + ε almost everywhere. This result was first stated by P. Erd˝ os in 1946 (see G. Tenenbaum [193], p. 344). Thus, q2 ≈ 1618, q3 ≈ 528 491 311 and q4 ≈ 514 843 556 263 450 564 886 528.